4,583 research outputs found
Optimization under Uncertainty in the Era of Big Data and Deep Learning: When Machine Learning Meets Mathematical Programming
This paper reviews recent advances in the field of optimization under
uncertainty via a modern data lens, highlights key research challenges and
promise of data-driven optimization that organically integrates machine
learning and mathematical programming for decision-making under uncertainty,
and identifies potential research opportunities. A brief review of classical
mathematical programming techniques for hedging against uncertainty is first
presented, along with their wide spectrum of applications in Process Systems
Engineering. A comprehensive review and classification of the relevant
publications on data-driven distributionally robust optimization, data-driven
chance constrained program, data-driven robust optimization, and data-driven
scenario-based optimization is then presented. This paper also identifies
fertile avenues for future research that focuses on a closed-loop data-driven
optimization framework, which allows the feedback from mathematical programming
to machine learning, as well as scenario-based optimization leveraging the
power of deep learning techniques. Perspectives on online learning-based
data-driven multistage optimization with a learning-while-optimizing scheme is
presented
OptNet: Differentiable Optimization as a Layer in Neural Networks
This paper presents OptNet, a network architecture that integrates
optimization problems (here, specifically in the form of quadratic programs) as
individual layers in larger end-to-end trainable deep networks. These layers
encode constraints and complex dependencies between the hidden states that
traditional convolutional and fully-connected layers often cannot capture. In
this paper, we explore the foundations for such an architecture: we show how
techniques from sensitivity analysis, bilevel optimization, and implicit
differentiation can be used to exactly differentiate through these layers and
with respect to layer parameters; we develop a highly efficient solver for
these layers that exploits fast GPU-based batch solves within a primal-dual
interior point method, and which provides backpropagation gradients with
virtually no additional cost on top of the solve; and we highlight the
application of these approaches in several problems. In one notable example, we
show that the method is capable of learning to play mini-Sudoku (4x4) given
just input and output games, with no a priori information about the rules of
the game; this highlights the ability of our architecture to learn hard
constraints better than other neural architectures.Comment: ICML 201
On Training and Evaluation of Neural Network Approaches for Model Predictive Control
The contribution of this paper is a framework for training and evaluation of
Model Predictive Control (MPC) implemented using constrained neural networks.
Recent studies have proposed to use neural networks with differentiable convex
optimization layers to implement model predictive controllers. The motivation
is to replace real-time optimization in safety critical feedback control
systems with learnt mappings in the form of neural networks with optimization
layers. Such mappings take as the input the state vector and predict the
control law as the output. The learning takes place using training data
generated from off-line MPC simulations. However, a general framework for
characterization of learning approaches in terms of both model validation and
efficient training data generation is lacking in literature. In this paper, we
take the first steps towards developing such a coherent framework. We discuss
how the learning problem has similarities with system identification, in
particular input design, model structure selection and model validation. We
consider the study of neural network architectures in PyTorch with the explicit
MPC constraints implemented as a differentiable optimization layer using CVXPY.
We propose an efficient approach of generating MPC input samples subject to the
MPC model constraints using a hit-and-run sampler. The corresponding true
outputs are generated by solving the MPC offline using OSOP. We propose
different metrics to validate the resulting approaches. Our study further aims
to explore the advantages of incorporating domain knowledge into the network
structure from a training and evaluation perspective. Different model
structures are numerically tested using the proposed framework in order to
obtain more insights in the properties of constrained neural networks based
MPC
Efficient representation and approximation of model predictive control laws via deep learning
We show that artificial neural networks with rectifier units as activation
functions can exactly represent the piecewise affine function that results from
the formulation of model predictive control of linear time-invariant systems.
The choice of deep neural networks is particularly interesting as they can
represent exponentially many more affine regions compared to networks with only
one hidden layer. We provide theoretical bounds on the minimum number of hidden
layers and neurons per layer that a neural network should have to exactly
represent a given model predictive control law.
The proposed approach has a strong potential as an approximation method of
predictive control laws, leading to better approximation quality and
significantly smaller memory requirements than previous approaches, as we
illustrate via simulation examples. We also suggest different alternatives to
correct or quantify the approximation error. Since the online evaluation of
neural networks is extremely simple, the approximated controllers can be
deployed on low-power embedded devices with small storage capacity, enabling
the implementation of advanced decision-making strategies for complex
cyber-physical systems with limited computing capabilities.Comment: 12 pages, 7 figure
Solving the L1 regularized least square problem via a box-constrained smooth minimization
In this paper, an equivalent smooth minimization for the L1 regularized least
square problem is proposed. The proposed problem is a convex box-constrained
smooth minimization which allows applying fast optimization methods to find its
solution. Further, it is investigated that the property "the dual of dual is
primal" holds for the L1 regularized least square problem. A solver for the
smooth problem is proposed, and its affinity to the proximal gradient is shown.
Finally, the experiments on L1 and total variation regularized problems are
performed, and the corresponding results are reported.Comment: 5 page
Differentiating through Log-Log Convex Programs
We show how to efficiently compute the derivative (when it exists) of the
solution map of log-log convex programs (LLCPs). These are nonconvex, nonsmooth
optimization problems with positive variables that become convex when the
variables, objective functions, and constraint functions are replaced with
their logs. We focus specifically on LLCPs generated by disciplined geometric
programming, a grammar consisting of a set of atomic functions with known
log-log curvature and a composition rule for combining them. We represent a
parametrized LLCP as the composition of a smooth transformation of parameters,
a convex optimization problem, and an exponential transformation of the convex
optimization problem's solution. The derivative of this composition can be
computed efficiently, using recently developed methods for differentiating
through convex optimization problems. We implement our method in CVXPY, a
Python-embedded modeling language and rewriting system for convex optimization.
In just a few lines of code, a user can specify a parametrized LLCP, solve it,
and evaluate the derivative or its adjoint at a vector. This makes it possible
to conduct sensitivity analyses of solutions, given perturbations to the
parameters, and to compute the gradient of a function of the solution with
respect to the parameters. We use the adjoint of the derivative to implement
differentiable log-log convex optimization layers in PyTorch and TensorFlow.
Finally, we present applications to designing queuing systems and fitting
structured prediction models.Comment: Fix some typo
LVIS: Learning from Value Function Intervals for Contact-Aware Robot Controllers
Guided policy search is a popular approach for training controllers for
high-dimensional systems, but it has a number of pitfalls. Non-convex
trajectory optimization has local minima, and non-uniqueness in the optimal
policy itself can mean that independently-optimized samples do not describe a
coherent policy from which to train. We introduce LVIS, which circumvents the
issue of local minima through global mixed-integer optimization and the issue
of non-uniqueness through learning the optimal value function (or cost-to-go)
rather than the optimal policy. To avoid the expense of solving the
mixed-integer programs to full global optimality, we instead solve them only
partially, extracting intervals containing the true cost-to-go from early
termination of the branch-and-bound algorithm. These interval samples are used
to weakly supervise the training of a neural net which approximates the true
cost-to-go. Online, we use that learned cost-to-go as the terminal cost of a
one-step model-predictive controller, which we solve via a small mixed-integer
optimization. We demonstrate the LVIS approach on a cart-pole system with walls
and a planar humanoid robot model and show that it can be applied to a
fundamentally hard problem in feedback control--control through contact.Comment: 7 pages, 8 figures. Submitted to the 2019 IEEE International
Conference on Robotics and Automation (ICRA 2019
Affine Multiplexing Networks: System Analysis, Learning, and Computation
We introduce a novel architecture and computational framework for formal,
automated analysis of systems with a broad set of nonlinearities in the
feedback loop, such as neural networks, vision controllers, switched systems,
and even simple programs. We call this computational structure an affine
multiplexing network (AMN). The architecture is based on interconnections of
two basic conceptual building blocks: multiplexers (), and affine
transformations (). When attached together appropriately, these
building blocks translate to conjunctions and disjunctions of affine
statements, resulting in an encoding of the network into satisfiability modulo
theory (SMT), mixed integer programming, and sequential convex optimization
solvers. We show how to formulate and verify system properties like stability
and robustness, how to compute margins, and how to verify performance through a
sequence of SMT queries. As illustration, we use the framework to verify closed
loop, possibly nonlinear dynamical systems that contain neural networks in the
loop, and hint at a number of extensions that can make AMNs a potent playground
for interfacing between machine learning, control, convex and nonconvex
optimization, and formal methods.Comment: 30 pages, 12 figure
LORM: Learning to Optimize for Resource Management in Wireless Networks with Few Training Samples
Effective resource management plays a pivotal role in wireless networks,
which, unfortunately, results in challenging mixed-integer nonlinear
programming (MINLP) problems in most cases. Machine learning-based methods have
recently emerged as a disruptive way to obtain near-optimal performance for
MINLPs with affordable computational complexity. There have been some attempts
in applying such methods to resource management in wireless networks, but these
attempts require huge amounts of training samples and lack the capability to
handle constrained problems. Furthermore, they suffer from severe performance
deterioration when the network parameters change, which commonly happens and is
referred to as the task mismatch problem. In this paper, to reduce the sample
complexity and address the feasibility issue, we propose a framework of
Learning to Optimize for Resource Management (LORM). Instead of the end-to-end
learning approach adopted in previous studies, LORM learns the optimal pruning
policy in the branch-and-bound algorithm for MINLPs via a sample-efficient
method, namely, imitation learning. To further address the task mismatch
problem, we develop a transfer learning method via self-imitation in LORM,
named LORM-TL, which can quickly adapt a pre-trained machine learning model to
the new task with only a few additional unlabeled training samples. Numerical
simulations will demonstrate that LORM outperforms specialized state-of-the-art
algorithms and achieves near-optimal performance, while achieving significant
speedup compared with the branch-and-bound algorithm. Moreover, LORM-TL, by
relying on a few unlabeled samples, achieves comparable performance with the
model trained from scratch with sufficient labeled samples.Comment: arXiv admin note: text overlap with arXiv:1811.0710
Machine learning approach to chance-constrained problems: An algorithm based on the stochastic gradient descent
We consider chance-constrained problems with discrete random distribution. We
aim for problems with a large number of scenarios. We propose a novel method
based on the stochastic gradient descent method which performs updates of the
decision variable based only on considering a few scenarios. We modify it to
handle the non-separable objective. Complexity analysis and a comparison with
the standard (batch) gradient descent method is provided. We give three
examples with non-convex data and show that our method provides a good solution
fast even when the number of scenarios is large
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